
It is bi-Hamiltonian [1]} and hence, integrable in this sense;

It is also integrable in the sense of having a Lax-pair, that is, it is the compatibility condition of the linear system
\(\begin{array}{lcl}\psi _{xx}&=&\displaystyle {\left(\zeta m-\zeta ^2\rho ^2+{4}\right)\psi },\\\\\psi _t&=&\displaystyle {\left({2\zeta }-u\right)\psi _x+{2}u_x\psi },\end{array}\) 
see [2]} for further details;

It is also of pseudo-spherical type [3]}, that is, the solutions of the system define one-forms that satisfy certain structure equations that implies on the existence of a two-dimensional Riemannian manifold with constant Gaussian curvature equals to \(-1\) ;

It is locally well-posed provided that \((u(0,x),\rho (0,x))\in H^{s}(\mathbb {R})\times H^{s-1}(\mathbb {R})\) , for \(s\ge 2\) , see [4]} and [5]}; its solutions, under some conditions, develop singularities in finite time in the form of wave breaking as well as it also possesses global solutions [5]}. These works are concerned with problems defined in the whole line, but there are also results about local and global existence of solutions as well as blow-up (wave breaking) for the system (REF ) provided that \(u(0,x)\)  and \(\rho (0,x)\)  are periodic solutions in \(H^{s}(\mathbb {S})\times H^{s-1}(\mathbb {S})\) , with \(s\ge 2\) , see [7]}, [8]};

It does not have non-trivial compactly supported solutions defined on the whole real line. In fact, Henry [4]} proved that if \(x\mapsto u(t,x)\)  is a \(C^2\)  function, compactly supported, and \((u,\rho )\)  solves (REF ), then \(u\)  and \(\rho \)  are identically zero.

System (REF ) can be seen as a geodesic flow on the tangent bundle of suitable infinite-dimensional Lie group [10]}, as well as the CH equation [11]}.

